View
218
Download
0
Category
Tags:
Preview:
Citation preview
CSE 337: Introduction to Medical Imaging
Lecture 6: X-Ray Computed Tomography
Klaus Mueller
Computer Science Department
Stony Brook University
Overview
Scanning:rotate source-detector pair around the patient
sinogram: a line for every angle
reconstruction routine
reconstructed cross-sectional slice
data
Early Beginnings
Linear tomographyonly line P1-P2 stays in focusall others appear blurred
Axial tomographyin principle, simulates the backprojection procedure used in current times
film
Current Technology
Principles derived by Godfrey Hounsfield for EMI• based on mathematics by A. Cormack• both received the Nobel Price in
medizine/physiology in 1979• technology is advanced to this day
Images:• size generally 512 x 512 pixels• values in Hounsfield units (HU)
in the range of –1000 to 1000
linear attenuation coefficient: air = -1000 water = 0 bone = 1000
• due to large dynamic range, windowing must be used to view an image
2
2
= 1000H O
H O
CT number (in HU)
CT Detectors
Scintillation crystal with photomultiplier tube (PMT)(scintillator: material that converts ionizing radiation into pulses of light) • high QE and response time• low packing density• PMT used only in the early CT scanners
Gas ionization chambers• replace PMT• X-rays cause ionization of gas molecules in chamber• ionization results in free electrons/ions • these drift to anode/cathode and yield a measurable electric signal• lower QE and response time than PMT systems, but higher packing
density
Scintillation crystals with photodiode• current technology (based on solid state or semiconductors)• photodiodes convert scintillations into measurable electric current• QE > 98% and very fast response time
Projection Coordinate System
The parallel-beam geometry at angle represents a new coordinate system (r,s) in which projection I(r) is acquired• rotation matrix R transforms coordinate system (x, y) to (r, s):
that is, all (x,y) points that fulfill r = x cos( ) + y sin()
are on the (ray) line L(r,) • RT is the inverse, mapping (r, s)
to (x, y)
s is the parametric variable along the (ray) line L(r,)
cos sin
sin cos
r x xR
s y y
cos sin
sin cosTx r r
Ry s s
Projection
Assuming a fixed angle , the measured intensity at detector position r is the integrated density along L(r,):
For a continuous energy spectrum:
But in practice, is is assumed that the X-rays are monochromatic
,
,
( , )
0
( cos sin , sin cos )
0
( ) Lr
Lr
x y ds
r s r s ds
I r I e
I e
,
( cos sin , sin cos )
0
0
( ) ( ) Lr
r s r s ds
I r I E e
Projection Profile
Each intensity profile I(r) is transformed to into an attenuation profile p(r):
• p(r) is zero for |r|>FOV/2 (FOV = Field of View, detector width)
• p(r) can be measured from (0, 2)• however, for parallel beam views (, 2) are redundant, so just need
to measure from (0, )
,0
( )( ) ln ( cos sin , sin cos )
rL
I rp r r s r s ds
I
I(r) p(r)
Sinogram
Stacking all projections (line integrals) yields the sinogram, a 2D dataset p(r,)
To illustrate, imagine an object that is a single point:• it then describes a sinusoid in p(r,):
projections point object sinogram
Radon Transform
The transformation of any function f(x,y) into p(r,) is called the Radon Transform
The Radon transform has the following properties:• p(r,) is periodic in with period 2
• p(r,) is symmetric in with period
( , ) { ( , )}
( cos sin , sin cos )
p r R f x y
f r s r s ds
( , ) ( , 2 )p r p r
( , ) ( , )p r p r
Sampling (1)
In practice, we only have a limited• number of views, M• number of detector samples, N• for example, M=1056, N=768
This gives rise to a discrete sinogram p(nr,m)• a matrix with M rows and N columnsr is the detector sampling distance is the rotation interval between subsquent views• assume also a beam of width s
Sampling theory will tell us how to choose these parameters for a given desired object resolution
rs
Sampling (2)
projection p(r)
spatial domain frequency domain
beam aperture s
smoothed projection
*
1 / s
sinc function
Sampling (3)
sampling at r
spatial domain frequency domain
sampled projection
smoothed projection
1 / r
.
1 / s
Limiting Aliasing
Aliasing within the sinogram lines (projection aliasing):• to limit aliasing, we must separate the aliases in the frequency
domain (at least coinciding the zero-crossings):
• thus, at least 2 samples per beam are required
Aliasing across the sinogram lines (angular aliasing):
1 2
2
sr
r s
max
max
max max
/ 2
for uniform sampling: / 2 2
k
Mk
kN
k k Nk M
M N
sinogram in the frequency domain
(2 projections with N=12 samples each are shown)
k
kmax=1/r
M: number of views, evenly distributed around the semi-circle
N: number of detector samples, give rise to N frequency domain
samples for each projection
Reconstruction: Concept
Given the sinogram p(r,) we want to recover the object described in (x,y) coordinates
Recall the early axial tomography method• basically it worked by subsequently “smearing”
the acquired p(r,) across a film plate• for a simple point we would get:
This is called backprojection:
0
( , ) { ( , )} ( cos sin , )b x y B p r p x y d
Backprojection: Illustration
Backprojection: Practical Considerations
A few issues remain for practical use of this theory:• we only have a finite set of M projections and a discrete array of N
pixels (xi, yj)
• to reconstruct a pixel (xi, yj) there may not be a ray p(rn,n) (detector sample) in the projection set
this requires interpolation (usually linear interpolation is used)
• the reconstructions obtained with the simple backprojection appear blurred (see previous slides)
1
( , ) { ( , )} ( cos sin , )M
i j n m i m j m mm
b x y B p r p x y
interpolation
detector samples
pixelray
To understand the blurring we need more theory the Fourier Slice Theorem or Central Slice Theorem• it states that the Fourier transform P(,k) of
a projection p(r,) is a line across the origin of the Fourier transform F(kx,ky) of function f(x,y)
A possible reconstruction procedure would then:• calculate the 1D FT of all projections p(rm,m), which gives rise to
F(kx,ky) sampled on a polar grid (see figure)• resample the polar grid into a cartesian grid (using interpolation)• perform inverse 2D FT to obtain the desired f(x,y) on a cartesian grid
However, there are two important observations:• interpolation in the frequency domain leads to artifacts • at the FT periphery the spectrum is only sparsely sampled
The Fourier Slice Theorem
polar grid
Filtered Backprojection: Concept
To account for the implications of these two observations, we modify the reconstruction procedure as follows:• filter the projections to compensate for the blurring• perform the interpolation in the spatial domain via backprojection
hence the name Filtered Backprojection
Filtering -- what follows is a more practical explanation (for formal proof see the book):• we need a way to equalize the contributions of all frequencies in the
FT’s polar grid• this can be done by multiplying each P(,k) by
a ramp function this way the magnitudes of the existing higher-frequency samples in each projection are scaled up to compensate for their lower amount
• the ramp is the appropriate scaling function since the sample density decreases linearly towards the FT’s periphery
ramp
Filtered Backprojection: Equation and Result
Recall the previous (blurred) backprojection illustration• now using the filtered projections:
2
0
( , ) ( ( , ) ) i krf x y P k k e dk d
ramp-filtering
inverse 1D Fourier transform p(r,) backprojection for all angles
not filtered filtered
1D Fourier transform of p(r,)
P(k,)
Filtered Backprojection: Illustration
Filters
There are various filters (for formulas see the book) :• all filters have large spatial extent convolution would be expensive• therefore the filtering is usually done in the frequency domain the
required two FT’s plus the multiplication by the filter function has lower complexity
Popular filters (for formulas see book):• Ram-Lak: original ramp filter limited to interval [±kmax]• Ram-Lak with Hanning/Hamming smoothing window: de-emphasizes
the higher spatial frequencies to reduce aliasing and noise
Ram-Lak
Hamming window
Windowed Ram-LakHanning window
Beam Geometry
The parallel-beam configuration is not practical• it requires a new source location for each ray
We’d rather get an image in “one shot”• the requires fan-beam acquisition
parallel-beam fan-beamcone-beam in 3D
Fan-Beam Mathematics (1)
Rewrite the parallel-beam equations into the fan-beam geometry
Recall:• filtering:
• backprojection:
• and combine:
0
( , ) *( , ) with cos sinf x y p r d r x y
/ 2
/ 2
*( , ) ( ', ) ( ') ' FOV
FOV
p r p r q r r dr
2 / 2
0 / 2
( , ) ( ', ) ( cos sin ') ' FOV
FOV
f x y p r q x y r dr d
Fan-Beam Mathematics (2)
• with change of variables:
• and after some further manipulations we get:
2 / 2
0 / 2
( , ) ( ', ) ( cos sin ') ' FOV
FOV
f x y p r q x y r dr d
= + sin r R
/ 22
20 / 2
1( , ) cos ( , ) ( ) ( )
( , ) 2sin( )
fan angle
fan angle
f x y R p q d dv x y
v(x,y) = distance from source
2. filter
the projection at
1. projection pre-weighting3. weighting during backprojection
Note: are the “new” r’, r
Remarks
In practice, need only fan-beam data in the angular range [-fan-angle/2, 180°+fan-angle/2]
So, reconstruction from fan-beam data involves• a pre-weighting of the projection data, depending on • a pre-weighting of the filter (here we used the spatial domain filters)• a backprojection along the fan-beam rays (interpolation as usual)• a weighting of the contributions at the reconstructed pixels,
depending on their distance v(x, y) from the source
Alternatively, one could also “rebin” the data into a parallel-beam configuration• however, this requires an additional interpolation since there is no
direct mapping into a uniform paralle-bealm configuration
Lastly, there are also iterative algorithms• these pose the reconstruction problem as a system of linear
equations• solution via iterative solves (more to come in the nuclear medicine
lectures)
Imaging in Three Dimensions
Sequential CT• advance table with patient after each
slice acquisition has been completed• stop-motion is time consuming and
also shakes the patient• the effective thickness of a slice, z, is equivalent to the beam width
s in 2D• similarly: we must acquire 2 slices per z to combat aliasing
Spiral (helical) CT• table translates as tube rotates around
the patient• very popular technique• fast and continuous• table feed (TF) = axial translation per
tube rotation• pitch = TF / z
z
z
Reconstruction From Spiral CT Data
Note: the table is advancing (z grows) while the tube rotates ( grows) • however,the reconstruction of a slice with constant z requires data from all
angles require some form of interpolation
• if TF=z/2 (see before), then a good pitch=(z/2) / z = 0.5 • since opposing rays (=[180°…360°]) have (roughly) the same information,
TF can double (and so can pitch = 1)• in practice, pitch is typically between 1 and 2• higher pitch lowers dose, scan-time, and reduces motion artifacts
sequential CT spiral CT
available data
interpolated
Spiral CT Reconstruction
3D Reconstruction From Cone-Beam Data
Most direct 3D scanning modality• uses a 2D detector• requires only one rotation around the patient to obtain all data (within
the limits of the cone angle)• reconstruction formula can be derived in similar ways than the fan
beam equation (uses various types of weightings as well)• a popular equation is that by Feldkamp-Davis-Kress• backprojection proceeds along cone-beam rays
Advantages• potentially very fast (since only
one rotation)• often used for 3D angiography
Downsides• sampling problems at the extremities
• reconstruction sampling rate
varies along z
Factors Determining Image Quality
Acquisition• focal spot, size of detector elements, table feed, interpolation method,
sample distance, and others
Reconstruction• reconstruction kernel (filter), interpolation process, voxel size
Noise• quantum noise: due to statistical nature of X-rays • increase of power reduces noise but increases dose• image noise also dependent on reconstruction algorithm,
interpolation filters, and interpolation methods• greater z reduces noise, but lowers axial resolution
Contrast• depends on a number of physical factors (X-ray spectrum, beam-
hardening, scatter)
Image Artifacts (1)
Normal phantom (simulated water with iron rod)
Adding noise to sinogram gives rise to streaks
Aliasing artifacts when the number of samples is too small (ringing at sharp edges)
Aliasing artifacts when the number of views is too small
Image Artifacts (2)
Normal phantom (plexiglas plate with three amalgam fillings)
Beam hardening artifacts• non-linearities in the polychromatic
beam attenuation (high opacities absorb too many low-energy photons and the high energy photons won’t absorb)
• attenuation is under-estimated
Scatter (attenuation of beam is under-estimated)• the larger the attenuation, the higher
the percentage of scatter
Image Artifacts (3)
Partial volume artifact • occurs when only part of the beam
goes across an opaque structure and is attenuated
• most severe at sharp edges
• calculated attenuation: -ln ( avg(I / I0))
• true attenuation: -avg ( ln(I / I0))
• will underestimate the attenuation
0 0- ln ( ( / ) (ln( / )) avg I I avg I I detector bin
I0 I
single pixel traversed by individual rays:
Image Artifacts (4)
Motion artifacts• rod moved during acquisition
Stair step artifacts: • the helical acquisition path becomes visible in the reconstruction:
Many artifacts combined:
Scanner Generations
Third generation most popular since detector geometry is simplest
• collimation is feasible which eliminates scattering artifacts
First
Second
Third
Fourth
Multislice CT
Nowadays (spiral) scanners are available that take up to 64 simultaneous slices (GE LightSpeed, Siemens, Phillips)• require cone-beam algorithms for
fully-3D reconstruction• exact cone-beam algorithms have
been recently developed
Multi-slice scanners enable faster scanning• recall cone-beam?• image lungs in 15s (one breath-hold)• perform dynamic reconstructions of the heart (using gating)• pick a certain phase of the heart cycle and reconstruct slabs in z
Single-slice Multi-slice
Exotic Scanners: Dynamic Spatial Reconstructor
Dynamic Spatial Reconstructor (DSR)• first fully 3D scanner, built in the 1980s by Richard Robb, Mayo
Clinic• 14 source-detector pairs rotating• acquires data for 240 cross-sections at 60 volume/s• 6 mm resolution (6 lp/cm)
Exotic Scanners: Electron Beam
Electron Beam Tomography (EBT)• developed by Imatron, Inc • currently 80 scanners in the world• no moving mechanical parts• ultra-fast (32 slices/s) and high resolution (1/4 mm)• can image beating heart at high resolution• also called cardiovascular CT CT (fifth generation CT)
CT: Final Remarks
Applications of CT• head/neck (brain, maxillofacial, inner ear, soft tissues of the neck)• thorax (lungs, chest wall, heart and great vessels)• urogenital tract (kidneys, adrenals, bladder, prostate, female genitals)• abdomen( gastrointestinal tract, liver, pancreas, spleen)• musceloskeletal system (bone, fractures, calcium studies, soft tissue
tumors, muscle tissue)
Biological effects and safety• radiation doses are relatively high in CT (effective dose in head CT is
2mSv, thorax 10mSv, abdomen 15 mSv, pelvis 5 mSv)• factor 10-100 higher than radiographic studies• proper maintenance of scanners a must
Future expectations• CT to remain preferred modality for imaging of the skeleton,
calcifications, the lungs, and the gastrointestinal tract• other application areas are expected to be replaced by MRI (see next
lectures)• low-dose CT and full cone-beam can be expected
Recommended